2.5 Applying the Power Rule

Learning objectives

—FUN-3.A—Calculate derivatives of familiar functions• FUN-3.A Calculate derivatives of familiar functions.• FUN-3.A.1 Direct application of the definition of the derivative and specific rules can be used to calculate the derivative for functions of the form f(x) = xʳ.• Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

A power function has the form $f(x)=x^r$ . Its derivative is found by moving the exponent $r$ in front as a coefficient, then subtracting $1$ from the exponent.

\frac{d}{dx}(x^r)=r x^{r-1}

Keep the base $x$ unchanged. Use the original exponent as the coefficient, calculate $r-1$ carefully, and then simplify only if the new form is clearer.

For $f(x)=x^{5/2}$ , $f'(x)=\frac{5}{2}x^{5/2-1}=\frac{5}{2}x^{3/2}.$ For $g(x)=x^{-3}$ , $g'(x)=-3x^{-4}=-\frac{3}{x^4}.$ The same rule handles positive, negative, and fractional powers; only the exponent arithmetic changes.

Do not multiply the exponent by $x$ and leave the exponent unchanged: the new exponent is always $r-1$ . Also preserve real-domain restrictions. For example, $x^{-3}$ and its derivative are not defined at $x=0$ .

Next Topic

—2.6 Derivative Rules: Constant, Sum, Difference, and Constant MultipleRead